Abstract
Sectorization means dividing a set of basic units into sectors or parts, a procedure that occurs in several contexts, such as political, health and school districting, social networks and sales territory or airspace assignment, to achieve some goal or to facilitate an activity. This presentation will focus on three main issues: Measures, a new approach to sectorization problems and an application in waste collection. When designing or comparing sectors different characteristics are usually taken into account. Some are commonly used, and they are related to the concepts of contiguity, equilibrium and compactness. These fundamental characteristics will be addressed, by defining new generic measures and by proposing a new measure, desirability, connected with the idea of preference. A new approach to sectorization inspired in Coulomb’s Law, which establishes a relation of force between electrically charged points, will be proposed. A charged point represents a small region with specific characteristics/values creating relations of attraction/repulsion with the others (two by two), proportional to the charges and inversely proportional to their distance. Finally, a real case about sectorization and vehicle routing in solid waste collection will be mentioned.

Abstract
Many models have been proposed for the location-allocation problem. In this study, based on sectorization concept, we propose a new single-objective model of this problem, in which, there is a set of customers to be assigned to distribution centres (DCs). In sectorization problems there are two important criteria as compactness and equilibrium, which can be defined as constraints as well as objective functions. In this study, the objective function is defined based on the equilibrium of distances in sectors. The concept of compactness is closely related to the accessibility of customers from DCs. As a new approach, instead of compactness, we define the accessibility of customers from DCs based on the covering radius concept. The interpretation of this definition in real life is explained. As another contribution, in the model, a method is used for the selection of DCs, and a comparison is made with another method from the literature, then the advantages of each are discussed. We generate benchmarks for the problem and we solve it with a solver available in Python’s Pulp library. Implemented codes are presented in brief.

Abstract
One of the most widely used methods in multi-objective optimization problems is the weighted sum method. However, in this method, defining the weights of objectives is always a challenge. Various methods have been suggested to achieve the weights, one of which is Shannon’s entropy method. In this study, a bi-objective model is introduced to solve the sectorization problem. As a solution method, the model is transformed into two single-objective ones. Also, the bi-objective model is solved for the case where the weights are equal to one. The gained three results from a benchmark are supposed as alternatives in a decision matrix. After the limitation of this approach appears, solutions from different benchmarks are added to the matrix. With Shannon’s entropy method, the weights of the objective functions are got from the decision matrix. The limitations of the approach and possible causes are discussed.

Abstract
In sectorization problems (SPs), a large area is divided into smaller regions for administrative purposes. SPs have applications in many fields. Since real-life problems are often dynamic, in this study, a new model for dynamic SP is proposed. In the problem, points are assigned to service centres and in this way sectors are formed. The sectors must be balanced in terms of distance and demand, which is defined in the objective function and constraints of the model. In the problem, in a certain time period, the coordinates and demands of some points change according to certain statistical distributions. A two-stage solution method is suggested for this problem. In the first stage, the expected values of coordinates and demands of the points are estimated by a Monte Carlo simulation, and in the second stage, the problem is solved like a deterministic optimization problem. The model is nonlinear, but after linearization, it is solved in Python’s Pulp library for benchmarks of different sizes and the results are discussed.

Abstract
Sectorization is the process of grouping a set of previously defined basic units (points or small areas) into a fixed number of sectors. Sectorization is also known in the literature as districting or territory design, and is usually performed to optimize one or more criteria regarding the geographic characteristics of the territory and the planning purposes of sectors. The most common criteria are equilibrium, compactness and contiguity, which can be measured in many ways. Sectorization is similar to clustering but with a different motivation. Both aggregate smaller units into groups. But, while clustering strives for inner similarity of data, sectorization aims at outer homogeneity [1]. In clustering, groups should be very different from each other, and similar points are classified in the same cluster. In sectorization, groups should be very similar to each other, and therefore very different points can be grouped in the same sector. We classify sectorization problems into four types: basic sectorization, sectorization with service centers, resectorization, and dynamic sectorization. A Decision Support System for Sectorization, D3S, is being developed to deal with these four types of problems. Multi-objective genetic algorithms were implemented in D3S using Python, and a user-friendly web interface was developed using Django. Several applications can be solved with D3S, such as political districting, sales territory design, delivery service zones, and assignment of fire stations and health services to the population.